Most information is passed on in the form of language. Therefore, research on how people use language to inform and misinform, and how this knowledge may be automatically extracted from large amounts of text is surely relevant. This survey provides first-hand experiences and a comprehensive review of rhetorical-level structure analysis for online deception detection. We systematically analyze how discourse structure, aligned or not with other approaches, is applied to automatic fake news and fake reviews detection on the web and social media. Moreover, we categorize discourse-tagged corpora along with results, hence offering a summary and accessible introductions to new researchers.